The conventional approach to the electronegativity equalization princi
ple suffers from a serious conceptual drawback that stems from discont
inuities in the first derivative of the electronic energy with respect
to the total charge of a molecule. A formalism that avoids the result
ing ill-defined atomic hardness matrices employs a simple definition o
f the atomic softness matrix sigma. The matrix sigma, which is a gener
alization of the Huckel atom-atom polarizability matrix, can be easily
and rigorously computed to arbitrary accuracy with any of the contemp
orary electronic structure methods. Properties of the atomic softness
matrix are reviewed and illustrated with several numerical examples in
volving the first-row hydrides and a linear polyyne.